[ExI] Intuitive Solution to Bell's Spaceship Paradox

Lee Corbin lcorbin at rawbw.com
Mon Aug 25 04:19:14 UTC 2008

    Intuitive Guide to SR and Bell's Spaceship Paradox

    Dialogue Concerning the Two Chief Spaceship Systems

Avoiding mathematics almost entirely, we proceed to

Step 1: Permanently disabuse oneself that simultaneity
        is real or frame-independent
Step 2: Despite step 1, know what constitutes a line
        (or a plane) of simultaneity in a reference frame
Step 3: Determine the *measurements* and logical 
        conclusions that the ground observers and
        the spaceship travelers must come to.
To this end, I am imagining a conversation I might
have with Galileo, were I instantly transported back to
1612 along with (a) a thorough knowledge of Italian and
the ability to articulately speak his dialect (b) a few
trinkets such as pocket calculators, wristwatches,
mechanical pencils and so forth with which to convince
him that I was as much ahead of him in history as he is
ahead of Aristotle, or more so.

The dialog is divided into four parts:

Part One:   Objective Simultaneity Exposed as a Fraud
Part Two:   Galileo Learns About Cartesian Coordinates
Part Three: Galileo Derives Properties of Special Relativity
Part Four:  Solving Bell's Spaceship Paradox in the 
                 Seventeenth Century

Part One:

Me: Good day, Signore Galileo. As I introduced myself to
    you last night, I promised you some fine discussion 
    for today, encompassing strange and wondrous future
GG: Good morning! Ah, to have a protracted philosophical
    discussion with someone from a future time is bound to
    be most enlightening, one way or another. Please proceed.
Me: As will not surprise you, you are quite right in
    your conclusions concerning the relativity of motion.
    But there are some incredible further consequences
    that can be deduced that took a full three centuries
    after your time to develop!
GG: I'm eager to hear of them. Pray continue.
Me: The first effect that I will convince you of is that
    it really is not possible to suppose that in any
    absolute sense two things can take place at exactly
    the same time, if there is any spatial distance
    whatever between them.
GG: Unbelievable! But surely one must happen before the
    other, or *else* they occur at the same time?
Me: First, we discovered in the 1880s the astonishing
    fact:  the speed of light is finite and even constant, no
    matter who is measuring it, whether it be a person in
    motion or one at rest---or of two people, as you would
    put it, who are in relative motion to one another. All
    the measurements will yield precisely the same velocity.
GG: How could this possibly be? 
Me: From the year 1880 or so until 1950, the finest philosophers
    in the entire world were just as baffled as you now find
    yourself to be, and even after that many went to their
    graves unwilling to admit that it could really be true,
    so outlandish and so offensive to common sense were the
    implications. Perhaps you can think of what some of those
    consequences might be.
At this point, Galileo shuts his eyes, starts muttering to
himself, and after three for four minutes---the equal of
you or me doing so for an hour---says:

GG: If I have understood the facts as you state them, and
    if these are indeed unimpeachable experimental results,
    then it must follow that the means of measuring this
    velocity of light, or the assumptions that go into 
    doing such, are faulty in some fundamental and most
    peculiar manner. Now if someone is to say that an 
    object, or even a ray of light, is approaching him
    or passing him by at a certain velocity, and this
    computation is at odds with another witness's, then
    from the standpoint of one witness, the other is 
    making an error either in his measurement of distance
    or in his measurement of time, because velocity is
    simply the ratio of the two! And, though I don't see
    how this could be true, it follows inescapably from
    what you have said.
Me: Yes, you have put your finger on the precise point.
    Either the other person's measurement of how much
    time has passed as the object moved from A to B,
    or the other person's measurement of what the actual
    distance is between A and B is---from our point of 
    view---faulty. So let's say that there is one specific
    location, take the middle of the Piazza della Signoria
    in Florence for one, and another specific location,
    take the Leaning Tower in Pisa, and let there be a
    cannonball that travels at uniform velocity between the
    two. Which will you have it, Signore, that the person
    traveling alongside the cannon ball no longer measures
    the distance between Florence and Pisa as do we, or 
    that his computation of the elapsed time is confounded?
GG: (Galileo pauses and thinks for a minute) Ah, but
    one must entail the other! For if as you say, were we
    to erect a giant mirror at the Tower and from the distant
    Piazza, and send a beam of light there and have it 
    reflect back to us, we must measure a different time
    interval than someone moving relative to us, I can
    intuitively feel that we would encounter a contradiction
    should we assume either that the distance he now measures
    between A and B has changed and the time remains the same
    or the other way around. I believe that I could show this
    mathematically. Let's see...
Me: Wow, I don't know how you did that, and I still don't quite
    see it myself, but it happens that you are quite right! For
    now, please, let us delay your calculations for the benefit
    of whosoever might be reading this dialog at some future
    time. I think that all you have just succeeded in intuitively
    feeling---because, sir, you are certainly one *tremendous*
    natural philosopher---I say, I think all this can be expressed
    in simple words as follows.
    Consider a very long Venetian galley proceeding at a very
    fast clip in a strong wind and with the rowers doing their
    utmost. In the twentieth century there are small, well-known
    devices called "flashbulbs" that emit an extremely bright
    but nearly instantaneous flash of light in all directions,
    at which point a flashbulb is said to "used up" or "shot".
    Is it clear that if you and I had a flashbulb in our hands
    right here and now, that a spherical expanding shell of
    light would spread out---at the speed of light, of course
    ---with this point where we stand remaining at the center
    of the various spheres as seen over time?
GG: Nothing could be clearer. But wait... I believe I see that
    there may be a difficulty here, if what you were saying 
    about the constant---
Me: Yes, but please allow me. Let us suppose that along the banks
    of the canal in Venice, we have set up a system of very
    precise clocks which measure time with the utmost accuracy,
    and which have all been completely synchronized. This is
    possible because the clocks are not in motion with respect
    to one another, and none of the difficulties I mentioned
    before will---
GG: Yes, in fact, one could send light signals between them, 
    and when finally, after an adjustment period, each person
    standing next to his clock would receive simultaneously
    the hour signal from the clock to his left and the clock
    to his right, even though he would have sent his own hour
    signal already.
Me: Your speed of comprehension is truly staggering. But yes,
    it's exactly as you say. This is possible only because
    there is no relative motion---
GG: Yes, as you had explained before: on this point, every
    witness measures the same times that it takes light to
    travel between any given clock and the ones to the left
    or right.
Me: Now let's suppose that just as the front end of the
    train is passing a clock that reads 12:00 the back
    end of the train is passing---
GG: Train?
Me: Sorry, the front end of the galley has a flash bulb
    that explodes just as an adjacent clock hits 12:00
    and a flash bulb explodes simultaneously next to
    a clock at the back end of the moving galley. Now
    given what we have said concerning the expanding
    spheres of light, you will admit that a witness
    situated exactly half way between these two clocks
    will see the flashbulb explosions at precisely the
    same instant.
GG: Of course. But... ah! Now you are going to insert 
    some additional clocks, if I am not mistaken, and
    this new series of clocks will be on the moving galley!
Me: Very good! Precisely. Now if we---
GG: Ah, but the witness on the galley who is at the exact
    midpoint of the galley will *not* see the two signals
    at the same time! Ah ha! And this could be proved!
    Suppose that at the exact instant the flash of light
    from the front of the vessel is made to cause a cannon
    to fire in one direction, and the light from the stern
    of the vessel causes a different cannon to fire. Then
    we must logically have that the cannon set to receive
    the signal from the front of the galley will fire first!
Me: And?
GG: And so... the witness next to the two cannons at the
    center of the galley must calculate---must measure, as
    you would say---that the flashbulb at the front of the
    ship exploded first! Wait. I think that I can calculate
    an exact formula for this effect. Suppose that v is the
    velocity of the ship, and the velocity of light is...
Me: Not now, please. You are about to derive formulas
    discovered between 1880 and 1905 known variously as
    "the Lorentz contraction", the "time-dilation effect",
    and so on. But let us see how much further it's possible
    to go without mathematics as yet.
GG: But my belief is that nature is written in the language...
Me: Oh yes, oh yes, we know all about your views on that, but
    again, please: not now.
GG: Well, you have persuaded me absolutely that if your
    "facts" concerning the speed of light are true, then
    indeed there can be no absolute simultaneity. It would
    all depend on relative motion. If one witness claims
    that A and B occurred simultaneously, then a traveler
    moving relative to him and towards B will claim---or,
    excuse me---measure that B happened before A.
Me: Yes. Well, let's take a break, if you don't mind and
    you can give me a nice little tour of Padua.
    *            *            *

Part Two. Galileo Learns About Cartesian Coordinates

GG: As we walked and ate, I was thinking these things 
    through, and though I do have a visceral understanding,
    I think, it would be very, very hard to persuade anyone.
Me: Are you aware of Cartesian coordinates? No, I think
    that this came around 1637, quite some time from now.
    What Descartes proposed was to draw a picture of the
    trajectory of a moving object by utilizing two lines,
    one of them horizontal that we will call the distance
    axis (or x-axis) and one vertical, which we shall call
    the time axis (or t-axis). Then it would be a simple
    matter to plot a coordinate pair, the x distance
    indicating the position, and the t distance indicating
    the time, both of which describe an object at a certain
    point in its trajectory.
GG: A very simple idea. I've played around with such conceptions,
    though they don't seem to aid the understanding in any
    particular way.
Me: It was found that such diagrams make the understanding
    of these thorny issues about light and time actually much
    simpler. Now if we were to calibrate the horizontal line
    in feet, and the vertical line in seconds, you can see
    that while this would suffice perfectly for describing,
    oh, say our just completed walk around Padua. But when
    we talk of---
GG: Ah, when we talk of the velocity of light, then such
    calibration... would be almost useless.
Me: Indeed, you seem to anticipate my every move! Anyway,
    we eventually found it most convenient to use one unit
    of distance on the x-axis correspond visually to one
    unit of time on the t-axis so that the speed of light
    would make a half a right angle.
GG: You mean that on such a diagram, while the rate of
    progress of a man walking would be scarcely distinguishable
    from the x-axis itself, something else moving at the
    velocity of light would make a forty-five degree angle,
    and the beam of light would be said to advance as far
    to the right as it does upward, supposing that we send
    light from the left to the right.
Me: Just so. Now let us imagine another horizontal line 
    situated not at t=0, but at t=1. That is, a horizontal
    line which indicates all the possible space and time
    positions that are simultaneous with our clock as our
    clock reads 1 second.
GG: Very well. Of course, there is an entire network of such
    horizontal lines, each one corresponding to a different
    precisely measured time.
Me: We call any such line a "line of simultaneity", for the
    simple reason that all the clocks at those indicated
    positions in space and time have precisely the same readings.
GG: Provided that they are at rest with respect to our clock.
Me: Er, yes, that's is exactly correct. Now the next thing that
    we must do is to attempt a very peculiar description on
    this so-called spacetime diagram. We are going to try to
    draw a new line that will correspond to all the spacetime
    position/events that are measured as simultaneous by the
    witness on the galley!
GG: I see. Hmm. Let me think about this for a moment. Suppose
    that we bring ashore and assemble all the witnesses who
    were on the passing galley at the times of interest. Well,
    firstly, we shall have to draw upon our diagram the single   
    report we already have, namely that the voyager at the
    galley's midpoint measures the flashbulb explosion at the
    ship's bow to have preceded the flashbulb explosion
    at the ship's stern. But we need something that he would
    have measured to be simultaneous, if we are to make progress
    on your suggestion that we depict a line of simultaneity as
    he would measure it.
Me: Let us say that the galley is moving from left to right.
    Now let A indicate the back end of the galley on the diagram
    at the moment we measured the flashbulb explosion to have
    occurred, and let B indicate the position of the front end
    of the galley at that exact time.
GG: Then we will have A at the center of the diagram where
    you have the time axis and the space axis cross...
Me: We call that the origin of the Cartesian Coordinate diagram...
GG: and so we we should place B about here, letting the distance
    between them on the x-axis represent the length of the galley.
Me: What we require is two events, two flashbulb explosions that
    will be recorded as simultaneous by the person at the center
    of the galley---
GG: Or in fact anywhere along the galley, because we must allow
    that they have set up a network of synchronized clocks every
    bit as well calibrated as our network of clocks.
Me: Yes. So where should we mark A' and B', two possibly new
    points on the diagram that will correspond to flashbulb
    explosions that will be recorded as simultaneous by the
    galley's passengers?
GG: Well, very clearly, in order to compensate for the effect
    so far discussed, we must have the flashbulb explosion at
    the rear of the vessel take place first. This will neatly
    account for the extra time it takes---from our measurements
    ---to reach the center of the galley since the galley is in
    motion. For simplicity, let us have A' coincide with A
    at your "origin" of the coordinate system.
Me: And B'?
GG: B' will lie somewhere over here to the right, probably right
    above B, but at say, for convenience x=1, t=.5.
Me: As a matter of fact, B' does not quite lie directly above B,
    if you recall your initial insight that one of the effects
    of time and space must entail the other.

Galileo shuts his eyes and after a while says, 

GG: Hmm, I think that B' will lie above and slightly to the right
    of B! Is that right?
Me: Well, I have *no* idea how you saw that, but then I'm not
    Galileo Galilee, one of the greatest physicists of all time,
    but yes, you are correct according to all the books I have
    read, and according to my own calculations as I have learned
    from several places how to perform.
    But it doesn't matter now. The essential point is that we
    now have two points on the galley's "line of simultaneity":
    one at the origin, and one at the position x=1, t=.5.
    What other points will there be?
GG: Quite obviously if there were a parallel galley in motion
    along side our first galley, and it were twice as long, 
    then the effect would scale up, and a flashbulb explosion
    at x=2, t=1 would be measured as simultaneous with the ones
    at x=1, t=.5 and x=0, t=0. And so this is the "line of
    simultaneity" we obtain once we fill in all the intermediate
Me: Yes. Well, I need to take a break before getting to the 
    next important phenomenon to tell you about, called time
GG: Care for some wine?

    *               *               *

Part Three. Galileo Derives Properties of Special Relativity
Me: Very good. Now let's forget about the first galleys we
    were discussing and I would like you to imagine an
    immense galley that is twenty thousand feet long and
    one thousand feet high. That is, in particular, the
    distance from the floor to the ceiling of the hold
    is 1000 feet.
GG: A most impressive craft. You have in your time such
    enormous ships?
Me: No. Big, but not that big. The purpose here is to make
    very concrete your earlier intuition that the time
    measurements of someone moving relative to you are
    systematically distorted from your measurements.
    Suppose that on the floor of the hold there is a
    perfectly flat mirror, and an equally perfect mirror
    adorns the ceiling of the hold. This is what I call
    the "Walt Disney arrangement" because when I was a
    boy a famous artist and entertainer drew a succession
    of many fine diagrams and presented them in a dramatization
    that made all of this very clear.
    I can now tell you something about the velocity of light.
    It so happens that light travels one foot in about one
    billionth of a second! And so, a beam of light that 
    began at the bottom mirror would reach---
GG: In one millionth of a second it would strike the upper
    mirror and be reflected towards the bottom one, and this
    too would take just one millionth of a second. Light 
    indeed travels most rapidly, if I am to believe your
Me: Yes, quite. Let us use the word "microsecond" for this
    millionth of a second. Now everyone's measurements are
    in perfect agreement so long as the galley has not yet
    begun to move. One microsecond up, one microsecond down.
    But what will the measurements be when the vessel is
    in motion?
GG: Ah, let me see. There must indeed be some effect. As I
    wrote in my own book---well, it's not published yet, 
    but I intend to get around to that---a person in the
    hold of a vast ship cannot discern any motion, for
    in principle it could be he who is at rest and the
    docks and the distant mountains that are moving.
Me: It so happens that in your honor we call this the "Galilean
    Principle of Relativity". So just to be perfectly clear,
    what times of transit up and down do the travelers on the
    the galley measure for the motion of the light between
    the mirrors?
GG: As I said, one microsecond up and one microsecond down.
    But I see where you are going with this. We who are
    ashore must measure something rather different, because 
    to us the light is not moving straight up and straight
    down, but on diagonal paths. 
    (more muttering and then a long pause)
    Ah, incredible. But it seems inescapable. Their very
    *time* is proceeding at a rate less than our own!
    Unbelievable. This is truly travolgente [mind-boggling].
    The very passage of time itself is not the same for
    them as it is for us. Now let's see, since the path length
    according to their measurements is ct (the speed of light
    times one micro-second), and the path length for us is
    by the Pythagorean Theorem---
Me: Not now. Please, our future readers---
GG: must be the square root of one minus the ratio of the 
    velocity of movement to the speed of light, each
    multiplied by itself.
Me: Yes, you've done it again. But let's return to principle
    here, the most vital component of understanding, the
    engine of intuition. For in reality the mathematics is
    pretty useless without a certain feel, or grasp of what
    is going on. So---what could you *now* say about the
    travelers who accompany the cannon ball flying between
    the Piazza and the Leaning Tower?
GG: Their measurement of elapsed time for the journey will be 
    less than our measurement. Ah! But! Since they are moving
    at a known velocity---and by symmetry all our measurements,
    theirs and ours, agree on that---then... they... must...
    measure the distance between the two cities to be less than
    we do! Incredible! Have I not made a mistake in reasoning
Me: No, you have not. You are completely correct in every
    particular. Indeed, they will *measure* the distance
    between Florence and Pisa to be less than the "true"
    distance, if by true we mean---
GG: If by "true" we mean the distance as measured by the
    cities themselves or rather by the presumed stationary
    inhabitants of the cities!
Me: Precisely. In fact, every mile of the distance will be,
    according to their measurements, less than one mile!
    And if they measure a donkey they pass by, it will
    be measured to be length-contracted in the direction
    of motion by the same token.
GG: This is simply too travolgente. Come, let us rest for
    a while and let me consider all this.
Me: Well, next, we will consider the most travolgente
    example of them all, the Bell Spaceship Paradox!
    *              *              *

Part Four. Solving Bell's Spaceship Paradox in the Seventeenth Century

GG: (Some time later.) What, may I ask, is a spaceship?
Me: A hypothetical device mentioned in many fantastical tales
    of my time that in some ways resembles an ordinary ship,
    but enables one to travel to the planets or even the stars.
GG: You may know that an eccentric correspondent of mine,
    who has many crazy but ingenious ideas about tides
    and about the paths of the planets, has fashioned 
    precisely such a fantastical account of travel to the
Me: Oh yes, he's almost as famous in our time as you are.
GG: But he did not make use of your "spaceships". So what
    is the big paradox all about?
Me: Imagine two of these vessels at rest with respect to
    the Earth, pointed in the same direction, but at the
    outset separated by one mile. Next imagine they begin
    to accelerate, and they accelerate in identical motions
    according both to our measurements as seen from the
    Earth and also according to the directions of the owners
    that the captains of each are following. A number of
    troubling questions arises concerning what occurs to them.
GG: Such as?
Me: First, can we be completely sure that according to *our*
    measurements they remain at the same original distance
    from each other?
GG: Well, that is what you yourself postulated. But yes,
    if I were to arrange to have one iron cannonball dropped
    from the top of the Pisa Tower and at the exact same
    instant another one dropped from a point half way down
    the tower, since the acceleration of each is the same,
    then there is no question that by our measurements
    their trajectories would be congruent. And so they
    would remain at the same "half-tower" distance apart
    with which they started, until the lower one struck
    the ground.
Me: Ah, but what about witnesses that were dropped alongside
    them? What would they measure?
GG: That is a very complicated question, because the velocities
    are not constant, and I sense that many difficulties will
    present themselves. I'm sure it can be worked out, but I
    think that I would like to work with one of those spacetime
Me: You're getting ahead of me once more, Signore. Now, firstly,
    we know that according to our measurements, the lengths of
    each spaceship will contract, correct?
GG: That is so, because it would be entirely analogous to the
    distance between Florence and Pisa, or analogous to the
Me: A certain very well known professor of our own time dreamt
    up a very interesting device concerning these two spaceships.
    He imagined a piece of string tied to the stern of the
    foremost spaceship and tied to the bow of the rearmost one.
    Now as the spaceships become shorter according to our
    measurements (but not according to the measurements of
    the occupants!), will the strings break?
GG: Oh, I see the problem! On one account, the entire arrangement
    of spaceship+string+spaceship may be regarded as a single
    entity, and so by our measurements the string will not 
    break, and this is also the answer that I believe would
    be arrived at by the passengers. But on an equally good
    account, the spaceships---as you postulated and as seems
    so reasonable---must remain at the same distance from
    each other; and so the strings would break!
Me: So do you see any way to find out the fact of the matter?
    After all, either the strings break or they do not, and
    all the witnesses by day's done must agree on that!
GG: It is a little clearer to me that the distances between
    the spaceships must remain the same, from our measurements
    at least, perhaps because you began by postulating that,
    or nearly so. An account could probably be told by which
    the opposite conclusion would come first to mind. Yet
    there is something troubling about regarding the
    spaceship+sting+spaceship as a single object, though 
    I cannot put my finger on it.
Me: Didn't you once use a piece of string yourself to intellectually
    demolish some argument of Aristotle's?
GG: Quite so. He believed that bodies fall at a velocity
    corresponding to their masses. So I imagined what would
    happen if you took two slowly falling bodies, and attached
    them with a string. Do they truly constitute one body at
    this time? Not really, because the string can be made so
    insubstantial by degrees. But as we gradually convert
    the string to a strong cord, then to a fastener made of
    iron, are we to expect that each increase will bring about
    a larger velocity? It seemed highly unlikely to me, and
    when I tried such experiments, it proved that the connector
    had nothing to do with it. 
Me: So in this spaceship case, would the string break?
GG: I'm thinking. (pause) What concerns me is this. Yes, it's
    clear on one account that the string would break, but the
    chief argument for this outcome arises from measurements
    that *we* would make who were not on board the ships.
    What measurements would they make? Let's produce one of
    those spacetime diagrams.
Me: Very well. Now let me ask you, what shape of curve should
    we use to depict the trajectories of the two spaceships.
GG: That's very simple. I proved long ago that these will be
    the conic section known as the parabola. You see, on this
    diagram the spaceship begins moving---well, you know what
    I mean---upwards on the diagram because it's hardly 
    changing its x-coordinate at all. Now, as it picks up
    speed, its motion begins to tilt to the right, like so.
    The motion is parabolic because of the law of falling
    squares: if an object falls one unit in one unit of 
    time, then it will have fallen four units in two units
    of time, nine in three, and so on.
Me: I probably should let you just work that out, because
    you'll get the right answer so far as the string goes.
    But you may be interested to know that the actual
    motion is a different conic section: the hyperbola.
GG: What!? Why?
Me: It's very complicated as to exactly why it's a hyperbola,
    and it's been many decades since I worked that out for
    myself, but fortunately we don't need to know the exact
    shape in order to resolve the Bell Spaceship Paradox.
    Here is a hint, though. Remember that I said that all
    experimental results indicate that all travelers, whatever
    their velocity, will measure the speed of light to be
    the same?
GG: Yes. You treated that as an axiom, and as odd as it seems,
    something tells me that this indeed could be the case.
Me: So imagine someone in a spaceship that goes faster and
    faster. At what velocity will he get away from light
    that is shining on him from behind?
GG: Hmm, since he measures it to be always constant..., why,
    at no velocity relative to us will he be able to escape
    the impingement on his vessel of the light shining from
    behind! And yet, of course, at any instant, in according
    to his measurements he's not the one moving anyone, it's
    all the rest of us.
Me: So?
GG: Therefore, on a spacetime diagram his velocity would
    asymptotically approach that of light according to
    our measurements. And his time would dilate to almost
    nothing. Hmm. And the curve would approach the fixed
    angle of forty-five degrees. So perhaps it would be
    a hyperbola indeed.
Me: The trouble with a parabola being what?
GG: Were we to draw a parabola here, then eventually its
    slope would become so small that the velocity depicted
    would exceed any given quantity---and movement faster
    than light would be possible, which is forbidden by
    what we have said. But why a hyperbola, exactly?
Me: As I say, I don't remember. The argument is a bit subtle.
    But you'll probably figure it out not long after I retire.
    Anyway, so what we do know for a fact is that the spaceship
    curve begins in some sort of arc that leans more and more
    over to the right.
GG: Yes. Ultimately approaching the forty-five degree line.
Me: Now, let us consider two such trajectories, one for the
    spaceship in advance, and one for the one behind. How
    would these be drawn?
GG: Clearly they would be congruent geometrical figures, for
    the precise reason that if I arrange to have two cannonballs
    dropped from different heights from a tower at precisely
    the same instant, then their trajectories will describe
    entirely congruent figures. So one curve is an exact copy
    of the other, just shifted one unit to the right.
Me: All right, so I see you have drawn the two approximately
    hyperbolic curves. Now comes the ultimate trick. We
    superimpose on the diagram a line of simultaneity for
    one of the moving spaceships at some point on its
GG: Very well, such a line begins at the origin and cuts 
    through both spaceship trajectory curves.
Me: Well, let's study that line a bit. Would you say that
    it strikes each parabola or hyperbola, whatever, at
    the same inclination?
GG: Why, no. Since the two curves are congruent and the
    line of simultaneity has a slope, then it cuts the
    first curve at a lower point than it cuts the second
    curve. And it won't be at quite the same angle either.
    For the curve on the right is being cut at a point
    further up, and so makes a smaller angle.
Me: Yet this *is* a line of simultaneity for all the
    points on the line, and in particular for the two
    points where the two spaceships are.
GG: Yes.
Me: Now let's pause a moment and ask ourselves how
    the trajectory lines of two galleys appear, when
    one of the galleys is moving quite a bit faster
    than the other---but, for simplicity's sake---
    at constant velocities.
GG: They have different slopes. The slower moving galley
    might have a line on the spacetime diagram that is
    nearly vertical, whereas the more quickly moving
    vessel has a line that is much more steeply inclined.
Me: Exactly. So let's look again at these two key points
    where the line of simultaneity cuts the trajectories
    of the two spaceships. What can be said about the
    velocities of the spaceships at those two points?
GG: The instantaneous slope conveys the magnitude of
    the velocity. So we see that along the line of 
    simultaneity where it cuts the two spaceship trajectories,
    the rearmost spaceship is moving more slowly!
Me: Because the tangent to the curve is less at that point
    than at the other point where the line of simultaneity
    cuts, right?
GG: Yes, just so. The tangent is a good way of seeing the
    instantaneous velocity. Hmm, well, together with this
    Descartes' coordinate system, I might very well go
    invent derivatives if I have some time next week, and
    thereby anticipate some other rather clever fellows
    later on.
Me: Yes, but now, what will the measurements made by each
    navigator on the two spaceships yield?
GG: On the lead spaceship, the navigator will be forced to
    report to his captain that the trailing spaceship is
    moving at a slower velocity---and, oh, that makes
    perfect sense! No wonder the string breaks! In their
    instantaneous coordinate system, the spaceships are
    actually pulling apart. The trailing spaceship, in
    their coordinates but not ours, is falling behind
    more and more!
Me: Yes, and I think that that is the final explanation.
    All the loose ends seem tied up now.

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